Lecture VLSI Digital signal processing systems: Chapter 14 - Keshab K. Parhi

Chapter 14: Redundant
Arithmetic
Keshab K. Parhi


• A non-redundant radix-r number has digits from
the set{0, 1, … , r - 1} and all numbers can be
represented in a unique way.
• A radix-r redundant signed-digit number system
is based on digit set S ≡ {-β, -(β - 1), … , -1, 0, 1, …
,α}, where, 1 ≤ β, α ≤ r - 1.
• The digit set S contains more than r values ⇒
multiple representations for any number in signed
digit format. Hence, the name redundant.
• A symmetric signed digit has α = β.
• Carry-free addition is an attractive property of
redundant signed-digit numbers. This allows most
significant digit (msd) first redundant arithmetic,
also called on-line arithmetic.
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Redundant Number Representations
• A symmetric signed-digit representation uses the digit set
D<r.α> = {-α, …, -1, 0, 1, …, α}, where r is the radix and α the
largest digit in the set. A number in this representation is
written as :
X<r. α> = xW-1.xW-2.xW-3…x0 = ∑ xW-1- iri
The sign of the number is given by the sign of the most

significant non-zero digit.
Digit Set D<r.α>

α

Redundancy Factor ρ

Incomplete

< (r – 1)/2

<½

Complete but non-redundant

= (r – 1)/2

=½

Redundant

≥ r/2

>½

Minimally redundant

= r/2

> ½ and < 1

Maximally redundant

=r–1

=1

Over-redundant

>r-1

>1

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Hybrid Radix-2 Addition
S<2.1> = X<2.1> + Y
where, X<r.α> = xW-1.xW-2xW-3…x0 , Y = yW-1.yW-2yW-3 …y0. The
addition is carried out in two steps :
1. The 1st step is carried out in parallel for all the bit positions.
An intermediate sum pi = xi + yi is computed, which lies in the
range {1, 0, 1, 2}. The addition is expressed as:
xi + yi = 2ti + ui,
where ti is the transfer digit and has value 0 or 1, and is
denoted as ti+ ; ui is the interim sum and has value either 1 or
0 and is denoted as -ui-. t-1 is assigned the value of 0.
2. The sum digits si are formed as follows:

si = ti-1+ - ui-

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Digit

Radix 2 Digit Set

Binary Code

xi

{1, 0, 1}

yi

{0, 1}

xi + - xi -

pi = xi + yi

{1, 0, 1, 2}

ui

{1, 0}

ti

{0, 1}

si = ui + ti-1

{1, 0, 1}

yi+

2ti + ui
-uiti+

si+ - si-

Eight-digit hybrid radix-2 adder
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Digit-serial adder formed by folding

LSD-first adder

MSD-first adder

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Hybrid Radix-2 Subtraction
S<2.1> = X<2.1> - Y
where, X<r.α> = xW-1.xW-2xW-3…x0 , Y = yW-1.yW-2yW-3 …y0. The
addition is carried out in two steps :
1. The 1st step is carried out in parallel for all the bit positions.
An intermediate difference pi = xi - yi is computed, which lies
in the range {2, 1, 0, 1}. The addition is expressed as:
xi - yi = 2ti + ui,
where ti is the transfer digit and has value 1 or 0, and is
denoted as -ti- ; ui is the interim sum and has value either 0
or 1 and is denoted as ui+ . t-1 is assigned the value of 0.
2. The sum digits si are formed as follows:
si = -ti-1- + ui+

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Digit

Radix 2 Digit Set

Binary Code

xi

{1, 0, 1}

yi

{0, 1}

xi + - xi -

pi = xi – yi

{2, 1, 0, 1}

ui

{0, 1}

ti

{1, 0}

si = ui + ti-1

{1, 0, 1}

yi-

2ti + ui
ui+

-ti-

si+ - si-

Eight-digit hybrid radix-2 subtractor
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Hybrid Radix-2 Addition/Subtraction

Hybrid radix-2 adder/subtractor (A/S = 1 for addition and
A/S = 0 for subtraction)
•This is possible if one of the operands is in radix-r complement
representation. Hybrid subtraction is carried out by hybrid
addition where the 2’s complement of the subtrahend is added
to the minuend and the carry-out from the most significant
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14
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position
is discarded.


•
•
•

•
•

Signed Binary Digit (SBD) Addition/Subtraction
Y<r.α> = Y + - Y -, is a signed digit number, where Y+
and Y- are from the digit set {0, 1, … , α}.
A signed digit number is thus subtraction of 2
unsigned conventional numbers.
Signed addition is given by:
S<r.α> = X<r.α> + Y<r.α> = X<r.α> + Y + - Y -,
⇒ S1<r.α> = X<r.α> + Y+,
S<r.α> = S1<r.α> - YDigit serial SBD adders can be derived by folding
the digit parallel adders in both lsd-first and msdfirst modes.
LSD-first adders have zero latency and msd-first
adders have latency of 2 clock cycles.

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Chap. 14

(a) Signed binary digit adder/subtractor
(b) Definition of the switching box

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Digit serial SBD redundant adders. (a) LSD-first adder
(b) msd-first adder
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Maximally Redundant Hybrid Radix-4 Addition
(MRHY4A)

• Maximally redundant numbers are based on digit set D <4.3>.
S<4.3> = X<4.3> - Y4
• The first step computes:
xi + yi = 4ti + ui
Replacing the respective binary codes from the table the
following is obtained :
(2xi+2 - 2xi-2 + 2yi+2) + xi+ - xi- + yi+ = 4ti+ + 2ui+2 - 2ui-2 - uiA MRHY4A cell consisting of two PPM adders is used to
compute the above.
• Step 2 computes computes si = ti-1 + ui. Replacing si, ui, and ti-1
by corresponding binary codes leads to si+2 = ui+2, si-2 = ui-2,
si+=ti-1+ and si- = ui-.

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Digit

Radix 4 Digit Set

Binary Code

xi

{3, 2, 1, 0, 1, 2, 3}

yi

{0, 1, 2, 3}

2xi+2 – 2xi-2 + xi+ - xi-

pi = xi + yi

{3, 2, 1, 0, 1, 2, 3, 4, 5, 6}

ui

{3, 2, 1, 0, 1, 2}

2ui+2 – 2ui-2 - ui-

ti

{0, 1}

ti+

si = ui + ti-1

{3, 2, 1, 0, 1, 2, 3}

2yi+2 + yi+
4ti + ui

2si+2 - 2si-2 + si+ - si-

Digit sets involved in Maximally Redundant
Hybrid Radix-4 Addition

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MRHY4A adder cell

Four-digit MRHY4A
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Minimally Redundant Hybrid Radix-4 Addition
(mrHY4A)

• Minimally redundant numbers are based on digit set D<4.2>.
S<4.2> = X<4.2> - Y4
• The first step computes:
xi + yi = 4ti + ui
Replacing the respective binary codes from the table the
following is obtained :

(- 2xi-2 + 2yi+2) + (xi+ + xi++ + yi+) = 4ti+ - 2ui-2 + ui+
A mrHY4A cell consisting of one PPM adder and a full adder
is used to compute the above.
• Step 2 computes computes si = ti-1 + ui. Replacing si, ui, and ti-1
by corresponding binary codes leads to si-2 = ui-2, si++ = ti-1+
and si+ = ui+.

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Digit

Radix 4 Digit Set

Binary Code

xi

{2, 1, 0, 1, 2}

yi

{0, 1, 2, 3}

– 2xi-2 + xi+ + xi++

pi = xi + yi

{2, 1, 0, 1, 2, 3, 4, 5}

ui

{2, 1, 0, 1}

2ui+2 – 2ui-2 - ui-

ti

{0, 1}

ti+

si = ui + ti-1

{2, 1, 0, 1, 2}

2yi+2 + yi+
4ti + ui

2si-2 + si+ + si++

Digit sets involved in Minimally Redundant
Hybrid Radix-4 Addition

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mrHY4A adder cell

Four-digit mrHY4A
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Non-redundant to Redundant Conversion
• Radix-2 Representation : A non-redundant number
X = x3.x2.x1.x0 can be converted to a redundant
number Y = y3.y2.y1.y0, where each digit yi is
encoded as yi+ and yi- as shown below:

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• Radix-4 representation :
– radix-4 maximally redundant number: X is a

radix-4 complement number, whose digits xi are encoded
using 2 wires as xi = 2xi+2 + xi+. Its corresponding
maximally redundant number Y is encoded using
yi = 2yi+2 - 2yi-2 + yi+ - yi-. The sign digit x3 can take values
-3, -2, -1 or 0, and is encoded using x3 = -2x3-2 - x3-.

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– radix-4 minimally redundant number: X is a radix4 complement number, whose digits xi are encoded using
2 wires as xi = 2xi+2 + xi+ . Its corresponding minimally
redundant number Y is encoded using yi = -2yi-2 + yi+ + yi++.
To convert radix-r number x to redundant number y<r.α>,
the digits in the range [α, r - 1] are encoded using a
transfer digit 1 and a corresponding digit xi - r where xi
is the ith digit of x. Thus,
2xi+2 + xi+ = 4xi+2 - 2xi+2 + xi+
= yi+1++ - 2yi-2 + yi+

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